Crystal structure and peculiarities of microwave parameters of Co1−xNixFe2O4 nano spinel ferrites

Nanosized spinel ferrites Co1−xNixFe2O4 (where x = 0.0–1.0) or CNFO have been produced using a chemical method. The crystal structure's characteristics have been determined through the utilization of X-ray diffraction (XRD). It has been demonstrated that all samples have a single phase with cubic syngony (space group Fd3̄m). The lattice parameter and unit cell volume behavior correlate well with the average ionic radii of Co2+ and Ni2+ ions and their coordination numbers. Thus, an increase in the Ni2+ content from x = 0.0 to x = 1.0 leads to a decrease in the lattice parameter (from 8.3805 to 8.3316 Å) and unit cell volume (from 58.86 to 57.83 Å3). Elastic properties have been investigated using Fourier transform infrared (FTIR) analysis. The peculiarities of the microwave properties have been analyzed by the measured S-parameters in the range of 8–18 GHz. It was assumed that the energy losses due to reflection are a combination of electrical and magnetic losses due to polarization processes (dipole polarization) and magnetization reversal processes in the region of inter-resonant processes. A significant attenuation of the reflected wave energy (−10 … −21.8 dB) opens broad prospects for practical applications.


Introduction
][8][9][10] The practical importance of such studies is due to the prospects for the wide use of multicomponent oxide systems in medicine, electronics, industry, ecology, etc.2][13][14][15] Ferrites contain iron oxide (Fe 2 O 3 ) as the main component, which endowes them with excellent magnetic properties.Depending on the formed crystal structure, there are four main groups of ferrites: spinels, garnets, orthoferrites, and hexaferrites.
The general formula for spinel ferrites is AB 2 O 4 , where A is a divalent ion (for example, Co 2+ , Ni 2+ , Cu 2+ , Mn 2+ , etc.) and B is a trivalent ion (usually Fe 3+ ).7][18] The distribution of cations between tetrahedral (A) and octahedral (B) positions has a signicant impact on the crystal structure, microwave, and magnetic properties of spinel ferrites. 19,20This aspect makes it possible to use spinel ferrites in ecology and biomedicine, electronics, and industry for various purposes, including targeted drug delivery, 21 MRI, 22 battery cathode materials, 23 longitudinal recording media, 24 and anodes, 25 gas sensors, 26 antennas, 27 microelectronics. 28errites based on nickel ions have outstanding magnetic and electrical characteristics (low eddy current loss and coercivity, high magnetic permeability and mechanical hardness, extremely high electrical resistivity, and high operating frequency).Therefore, they are subjected to diversied research.The potential applications of nickel-based ferrites are numerous and promising, such as high-density and high-

Synthesis
The following steps have been used to synthesize CNFO nanoparticles using the citrate-nitrate auto combustion approach: initially, a combination of chemical reagents including Co(NO 3 ) 2 $6H 2 O ($98.5% purity, sourced from Qualikems), Ni(NO 3 ) 2 $6H 2 O ($98% purity, sourced from Advent Chembio Pvt.Ltd), Fe(NO 3 ) 3 $9H 2 O ($98.5% purity, sourced from Qualikems), and C 6 H 8 O 7 ($99.5% purity, sourced from Oxford Lab Fine Chem LLP) have been adjusted in weights, such that the ratios of the used anhydrous citric acid to trivalent metal ions to and divalent metal ions were 3 : 2 : 1. [39][40][41] These nitrate salts were dissolved in distilled water and stirred for 15 minutes without heating to create a homogeneous mixture before adding the citric acid.Finally, ammonium hydroxide was added to the mix in drops while the stirring continued until the pH reached approximately 7.0. 42,43Aer removing the magnet, the mixture was heated to about 120 °C for three hours.A formed viscous gel has been self-ignited to give a ne and brown powder of the ferrite eventually.The material has been grained by agate mortar, then pressed to discs and sintered at 900 °C for 4 hours.Finally, the disc-shaped CNFO samples were ground again and prepared for characterization.The following formula can characterize the synthesis of the studied CNFO compositions: shows the preparation method of CNFO nano ferrites.Center and Measurements of Tanta University.CNFO microstructure was investigated with SEM (Zeiss EVO 10, Oberkochen, Germany).The chemical composition of nanosized CNFO was studied using EDXS (AZtecLive Advanced Ultim Max 40 detector, Oxford Instruments, Bognor Regis, UK).The average particle size of CoFe 2 O 4 was estimated by TEM (JEM-2100 instrument at the National Research Center in Cairo).The vector network analyzer (Agilent) has investigated the microwave properties of ferrites.[46][47]

Results and discussion
Crystal structure and elastic properties These results benet the selected synthesis and graing processes, which are expected to achieve better material performance for intended applications.The crystallinity of CNFO composites is also conrmed by the fact that the detected XRD peaks are sharp and narrow.Moreover, the main characteristic peaks of the CNFO compounds undergo a slight shi towards larger 2q angles with an increase in the Ni 2+ concentration.Such a shi can be associated with a change in dspacing, determined by the lattice parameter and volume, and is explained by the fact that the nickel ion has smaller ionic radii (0.63 Å) with regard to the cobalt ion (0.74 Å). 48This also conrms that Ni 2+ is a good substitute for Co 2+ in CNFO.The crystal size was determined using the Scherrer formula: where l is the X-ray wavelength (= 1.54 Å), k is a constant, and its value depends on the crystallite shape (= 0.89 for cubic crystals), b denotes the FWHM (full width at half maximum) of the main peak representing the planes (311), q represents the Bragg angle.
where d is the interplanar spacing, and it can be determined using Bragg's equation as follows: The following equation was used to determine the X-ray density for CNFO samples (D x ): where M represents the molecular weight of the sample, N A is Avogadro's number, and a 3 represents the unit cell volume.Furthermore, jump length (the distance between magnetic ions) in the tetrahedral A-site L A-A (Å), octahedral B-site L B-B (Å), and shared sites L A-B (Å) have been estimated using the following equations: 39,49 L AÀA ¼ a ffiffi ffi 3 p

4
; The estimation of dislocation density, dened as the number of dislocation lines per unit volume of crystal, represented by the symbol d, can be achieved through the utilization of the following equation: The estimation of the mean ionic radius per molecule for the tetrahedral and octahedral sites, denoted as r A and r B, respectively, has been calculated by utilizing the cation distribution for each composition and using the following relations: where f represents the fractional concentration, r refers to the ionic radius of the respective cation on the respective site.
The theoretical value of the lattice parameter a Th was calculated for all CNFO compositions from the following formula: 52,53 a Th = 1.5396 where r O is the oxygen ion radius.
Fig. 2 displays the concentration dependencies of the main structural parameters of CNFO (x = 0.0-1.0)obtained from the X-ray diffraction (XRD) data.Some parameters are listed in Table 1.The crystal size varies from 46.61 to 55.41 nm, as demonstrated in Fig. 2c.A monotonic increase is observed with an increase in Ni 2+ content, which is expected at a sintering temperature of 900 °C for 4 hours.This can be explained by decreasing lattice strain, which usually contributes to increased crystallite size. 53s the nickel content increases, a bigger Co 2+ ion (0.745 Å at the octahedral site and 0.580 Å at the tetrahedral site) is substituted by a smaller Ni 2+ ion (0.690 Å at the octahedral site and 0.550 Å at tetrahedral site) in the CNFO ferrites 54 Thus, there is a linear decrease in the lattice parameter and unit cell volume, with increasing Ni 2+ concentration, following the trend expected from Vegard's rule. 55The calculated values of theoretical lattice parameters correlate well with the established (a Th ) and (a) values, which conrms the reliability of the proposed distribution of cations for the CNFO system.A small difference between the calculated lattice parameters (a Th ) and their experimental values (a) was expected due to the changes in the distribution of cations among all ions. 56Both cobalt and nickel ferrites crystallize into inverse spinels when Co 2+ and Ni 2+ cations prefer to occupy octahedral (B) positions.
In contrast, the Fe 3+ cations exhibit distribution across both octahedral (B) and tetrahedral (A) positions.This distribution can be inuenced by many factors (e.g., annealing temperature), which can cause a small amount of Co 2+ and Ni 2+ cations to move to (A) positions.Also, it is observed that r A and r B values show almost a decreasing trend with an increase in nickel concentration.These changes in r A and r B are associated with different distributions of cations in tetrahedral and octahedral positions.This means that the concentration of Co 2+ (Ni 2+ ) cations in the A-and B-positions increases (reduces) the cation redistribution and the migration of some of the Fe 3+ ions from the A-positions to the B-positions.Since cations' distribution strongly inuences ferrites' magnetic properties, the assumption made is also investigated, considering the obtained magnetic characteristics, which will be presented later in this study.
The observed reduction in the unit cell volume (lattice parameter (a)) exceeds the reduction in molecular weight.The X-ray density (D x ) increases with an increase in the nickel concentration since the atomic mass of Co 2+ (58.933AMU) is very close to that of Ni 2+ (58.693AMU).As a result of the formation of pores during disc-shaped pressing of CNFO samples and sintering processes at high temperatures, X-ray density values (D x ) are observed to be higher than the measured density values (D).The porosity of the CNFO system is low and tends to increase with increasing nickel concentration.Table 1 Structural parameters of the CNFO: ionic radius of tetrahedralr A (Å) and octahedral sitesr B (Å), theoretical lattice constanta Th (Å); interlattice and intralattice distance - This can be attributed to abnormal compaction and imperfection of the crystal structure, which depends on stoichiometry, synthesis method, heat treatment conditions, and the interplay between D and D x . 53As the concentration of Ni 2+ ions increases, the number of atomic defects, such as dislocation density (d) and strain (3), are observed to decrease, displaying an inverse correlation with the crystallite dimensions.This phenomenon aligns with the expected patterns.The contraction of lattice parameters is driven by a decrease in the proximity between magnetic ions, a process referenced in previous literature. 56his spatial shi results from the cation redistribution that occurs due to the substitution of Ni ions. 53onsequently, L A-A , L B-B , and L A-B values diminish with a growing concentration of nickel, which is consistent with theoretical predictions based on relevant equations.Observations reveal a clear pattern: the value of L A-A consistently surpasses L B-B .This suggests that the likelihood of electron transition between ions residing at A and B sites is lower than between ions located at the same B site. 55,56TIR spectra of CNFO (x = 0.0-1.0)samples are illustrated in Fig. 3.The vibrational bands located around 575-595 cm −1 and 375-395 cm −1 are attributed to the stretching vibrations of the tetrahedral and octahedral groups, respectively.These vibrational modes are commonly referred to as (n 1 ) and (n 2 ).The presence of these two prominent metal-oxygen absorption bands is a fundamental characteristic observed in all FTIR spectra of spinel ferrite nanoparticles documented in the literature. 55,57,58The bands observed at 1637 and 2922 cm The identiable band suggests the formation of hydrogen bonds amongst hydroxyl groups, providing evidence for the presence of either free or adsorbed water within the sample.A broad absorption band detected at 3469 cm −1 is ascribed to the stretching vibrations of the O-H bond in water molecules present within the interstitial spaces of the layers. 42,57Minute changes in the intensity and minor spectral shis in the two primary characteristic absorption bands of the ferrites are noted, as presented in Table 2 and Fig. 3.The reason for these observed anomalies may be correlated with factors such as changes in the effective atomic mass, bond lengths, force constants, and the electronegativity of the cations. 59The shi observed in the position of the absorption band, originating from the tetrahedral and octahedral sites, is linked to the force constant.This constant is directly proportional to the atomic number of the metal ions, the atomic number of the oxygen ions, and the length of the metal-oxygen bond, respectively.The force constants for the tetrahedral site, denoted as F 1 , and the octahedral site, denoted as F 2 , can be computed using the provided equations: where c represents the velocity of light in vacuum, while m represents the reduced mass of the system comprising oxygen and metal, which is equivalent to 2.061 × 10 −23 g.Table 2 consolidates the Fourier Transform Infrared (FTIR) data accrued from the analyzed samples.It is observed that the force constant values for F 1 surpass those for F 2 , a phenomenon that aligns with expectations.This discrepancy can be attributed to variations in the stretching of bands between n 1 and n 2 , intensied cationic interactions within the tetrahedral site, decreased interatomic distances, and the increased energy prerequisite for bond disruption. 60he modications in the force constants F 1 and F 2 can be linked to the cation redistribution within the tetrahedral and octahedral sites, which occurs in correlation with variations in grain sizes. 55heir elastic characteristics indicate the isotropy and homogeneity of materials.Together, XRD and IR spectral analyses make their estimation easier.The elastic stiffness constants of the spinel ferrite system can be calculated using the following equations: where F av represents the average force constant of F A and F B while s denotes the Poisson's ratio, which is dependent on porosity (P) given by: For solids exhibiting a cubic structure, the estimation of their elastic moduliencompassing the bulk modulus (B), Young's modulus (E), and the shear modulus, also known as rigidity modulus (G)can be conducted via the utilization of the computed stiffness constants.The corresponding equations used for these calculations are as follows: Moreover, the X-ray density (D x ) and the stiffness constant C 11 can be deployed to calculate the velocities of longitudinal and transverse elastic waves, represented by V l and V s , respectively.
The calculations are guided by the subsequent equations, as indicated in the ref.49 and 60: Finally, the Debye temperature Q D can be calculated using the formula: where h represents Plank's constant, k represents Boltzmann's constant, c is the speed of light, n av is the average value of wavenumbers for n 1 and n 2 at A and B-sites.
Table 2 presents all estimated values obtained from the calculations.The observed pattern of the stiffness constant C 11 escalating with increasing Ni content can be interpreted as a consequence of the fortication of interatomic bonds amongst distinct atoms within the spinel lattice.Concurrently, the stiffness constant C 12 displays a decrease, indicating a probable decline in the crystallinity of the samples with escalating Ni content.Both the longitudinal (V l ) and transverse (V s ) elastic wave velocities demonstrate an increase correlating with the rise in Ni content, a phenomenon attributable to an elevation in the average force constant.It is projected that V l values will surpass V s values due to the understanding that transverse waves necessitate lesser energy to instigate particle vibration perpendicular to the direction of wave propagation, in contrast to longitudinal waves that demand higher energy to stimulate particle vibration in the parallel direction. 61,62Poisson's ratio decreases as expected since the porosity values increase, as shown above in Table 2.It is well known that X-ray density affects elastic moduli. 63Therefore, since the values of Xray density increase with increasing nickel content, as shown in Table 2, the elastic moduli B, G, and E increase as functions of density.
It has been observed that the Debye temperature shows an upward trend with increasing the concentration of nickel.The observed trend in the Debye temperature can be attributed to the rise in the wave number of vibrational bands of the infrared spectra, as well as a rise in the stiffness of the samples. 61ig. 4 The TEM micrographs reveal the existence of nano-scale particles and numerous agglomerations, consequences of magnetic inter-particle interactions.The identied interplanar spacing for CoFe 2 O 4 is 0.2604 nm, corresponding to the (311) crystal plane's interplanar spacing.The histogram revealing the average particle size distribution is derived from the TEM images, facilitated by ImageJ soware, and involves the analysis of a total of 95 particles within the TEM image.The mean particle size is identied as 52.1 nm, which aligns with the average crystallite size determined through the Scherrer Table 2 FTIR absorption band values, force constants F n1 (dyne per cm) and F n2 (dyne per cm) at A-and B-sites, average force constant F av (dyne per cm), elastic stiffness constants C 11 (GPa) and C 12 (GPa), longitudinal V l (m s −1 ) and transverse V s (m s −1 ) wave velocities, bulk modulus B (GPa), rigidity modulus G (GPa), Young modulus E (GPa), Poisson's ratio (s), and Debye temperature Q D (K) of CNFO (x = 0.0-1.0)equation, leveraging X-ray Diffraction (XRD) data.The minor disparity between the average sizes determined via XRD and TEM arises from the fact that XRD measures crystallite size.At the same time, TEM gauges the dimensions of whole particles.
The SAED pattern reveals ring patterns, underscoring the crystalline nature of cobalt ferrite.
The surface topography and morphology of CNFO (x = 0.0-1.0)samples were investigated utilizing Scanning Electron Microscopy (SEM), with results illustrated in Fig. 5.
Analysis of the SEM imagery conrms that the samples possess a dense and homogenous structure, featuring ne spherical particles with irregularly oriented grain aggregations.The compositional analysis of the samples was carried out through Energy Dispersive X-ray Spectroscopy (EDXS).As demonstrated in Fig. 4, the ndings verify the existence of Co, Ni, Fe, and O, affirming the successful, uncontaminated synthesis of the samples, devoid of any unintended elements.Furthermore, the observed escalation in the intensity of nickel depicted in the graphs, corresponding with the rise in Ni content, implies its successful integration within the CoFe 2 O 4 ferrite structure.

Microwave parameters
Fig. 6 illustrates the frequency-dependent behavior of the permittivity for CNFO with (x = 0.0-1.0).The measurements were conducted within the frequency range of 8 to 18 gigahertz.
The values of the real and imaginary parts of the electric permittivity and magnetic permeability were calculated from the measured S-parameters.In Fig. 6a GHz, local minima are noted for all the samples.For both samples with x = 0.0 and 1.0, two clear minima are noted.While for the rest of the samples, there is blurring by local minima (oen with the formation of a wide plateau, as for x = 0.5).The amplitude of these minima correlates well with the nickel concentration.So, it can be noticed that the minimum value of the amplitude is typical for x = 0.0 (∼1.42), while the maximum value is noted for x = 1.0 (∼1.62).The presence of local minima may be due to polarization losses (Fig. 6c).Thus, a signicant reduction in the real part of the permittivity with an increase in the imaginary part (Fig. 6b demonstrates the presence of a peak in the frequency dependences of the imaginary part of the permittivity) may correspond to dipolar polarization.The polarization processes are attributed to the real part of the permittivity.
In contrast, the imaginary part is responsible for absorption due to electrical losses in the material.Relatively low values of the imaginary part of the permittivity indicate the absence of signicant absorption due to electrical losses.This behavior of the electrical permittivity can be observed due to a change in the electronic conguration of the A-cation (substitution of a Co 2+ ion by Ni 2+ ion).Thus, the Ni 2+ ion has the conguration of the outer electron shells 3d 6 4s 2 , while the conguration of Co 2+ is 3d 5 4s 2 .The presence of a large number of unpaired highly localized electrons (electrons that are not participating in the formation of chemical bonding and conductivity) in Co 2+ (3d 5 ) allows the formation of a larger dipole moment during polarization in the high-frequency region.This is reected in the larger amplitude of the local minimum on the frequency dependence of the real part of permittivity.With an increase in the Ni 2+ concentration (Ni 2+ ion with a smaller number of unpaired highly localized electronsonly 4 unpaired electrons per 3d 6 orbitals), the contribution to polarization losses decreases.Fig. 7 illustrates the frequency-dependent behavior of magnetic permeability.It is widely recognized that the magnetic  permeability's real part arises from the reversal processes of magnetization.Simultaneously, the magnetic permeability's imaginary part is accountable for losing the resonant naturethe domain boundaries resonance (DBR) and the natural ferromagnetic resonance (NFMR).The rst type of resonance (DBR) is associated with the intense absorption of electromagnetic radiation energy at frequencies corresponding to the internal frequencies of the domain wall's motion.DBR is noted at lower frequencies compared to the NFMR.The second type of resonance (NFMR) is caused by the intense absorption of electromagnetic energy at frequencies corresponding to the precession frequency of the magnetic moment of the electronic subsystem.When analyzing Fig. 7a, it can be noted that at frequencies up to 12 GHz, the minimum values (∼1.21-1.26) of the real part of the magnetic permeability are characterized by the x = 0 sample (pure cobalt ferrite).
All other samples have higher values (in the range of ∼1.45-1.99) of actual permeability.This can be revealed by the fact that cobalt ferrite has higher values of magnetocrystalline anisotropy and coercive force, which requires more energy for the magnetization reversal processes and makes it difficult for the electromagnetic ux to pass through the material (lower permeability values on frequency dependences).With an increase in the concentration of Ni 2+ ions, the values of the magnetocrystalline anisotropy and coercive force decrease, which increases the values of the magnetic permeability and results in an increase in the real part of the magnetic permeability.At frequencies above 12 GHz, the situation changes.The maximum values (reaching 4.0) characterize samples with the maximum concentration of cobalt (x = 0.0 and 0.3).This may be due to the inuence of the higher magnetic moment of the cobalt ion compared to the nickel ion.Relatively low values of the imaginary part of the permeability (Fig. 7b) and the absence of a clearly dened peak in the frequency dependences may indicate the absence of intense absorption due to magnetic losses of a resonant nature.
The minimum values of the imaginary part (∼0.06-0.28)are noted for the x = 0.0 sample (which can also be explained by the higher value of the magnetocrystalline anisotropy for pure cobalt ferrite).The maximum values of the imaginary part are noted (∼0.29-0.76)for the x = 1.0 sample.Let's analyze the frequency dependences of the real and imaginary parts of the magnetic permeability.We can conclude that the studied frequency range (8-18 GHz) is in the intermediate region between the two resonances.Fig. 7c shows the mechanism that explains the behavior of magnetic permeability with increasing frequency of electromagnetic radiation.A slight joint increase in both the real and imaginary parts of the magnetic permeability indicates the transition of the magnetic loss mechanism from domain wall resonance to natural ferromagnetic resonance.The energy losses in reection can be expected to be a combination of electrical and magnetic losses due to polarization processes (dipole polarization) and magnetization reversal processes in the region of inter-resonant processes.
Fig. 8 illustrates the frequency-dependent behavior of the reection coefficient (reection losses), representing the re-ected wave's energy losses.Negative values of the coefficient indicate the attenuation of the reected radiation.It should be noted that for all samples (except x = 1.0), signicant reection losses were noted (with a reection coefficient of more than −10 dB).For each sample (except x = 1.0), 2 broad peaks can be distinguished on the frequency dependences.Smaller peaks at frequencies above 13 GHz may be due to the combined energy losses due to magnetic and electrical losses.At the same time, peaks in the range of up to 13 GHz can result from multiple reection processes in the material itself and processes of superposition of the reected and incident waves.

Conclusions
CNFO or Co 1−x Ni x Fe 2 O 4 (x = 0.0-1.0)nanoparticles have been produced using a chemical method (citrate-nitrate auto combustion technique).All samples have been examined using XRD, which has conrmed the formation of the cubic spinel ferrite structure (space group Fd 3m) without any detectable impurities (single phase).The lattice parameter and unit cell volume behavior correlate well with the average ionic radii of Co 2+ and Ni 2+ ions and their coordination numbers.Thus, an increase in the Ni 2+ content from x = 0.0 to x = 1.0 leads to a decrease in the lattice parameter (from 8.3805 to 8.3316 Å) and unit cell volume (from 58.86 to 57.83 Å 3 ).The purity of the chemical composition (absence of undesirable impurities or phases) and the successful synthesis of nanosized spinel ferrites Co 1−x Ni x -Fe 2 O 4 have been conrmed by the EDXS data.The obtained compositions of the Co 1−x Ni x Fe 2 O 4 system are homogeneous and contain small spherical particles with unevenly oriented grains in the form of aggregates, which was established during the microstructure analysis.Through the analysis of TEM images, the mean particle size was determined, and these values exhibited good agreement with the average crystallite size derived from XRD data using the Scherrer equation.Fourier Transform Infrared (FTIR) studies indicated the formation of hydrogen bonds amongst hydroxyl groups in Co 1−x Ni x Fe 2 O 4 spinel ferrites, suggesting the existence of either adsorbed or free water within the samples.The observed variation in the force constants of the tetrahedral and octahedral sites was elucidated through the cation redistribution that occurred between these positions, a process triggered by alterations in grain size.Furthermore, the characteristics of the microwave properties were examined based on the S-parameters measured within the frequency range of 8-18 GHz.Aer analyzing the frequency dispersions of the permittivity and permeability, the main mechanisms of the EMR interaction with condensed matter in the high-frequency range were discussed.It was assumed that the energy losses due to reection are a combination of electrical and magnetic losses due to polarization processes (dipole polarization) and magnetization reversal processes in the region of inter-resonant processes.A signicant attenuation of the reected wave energy (−10 .−21.8 dB) opens broad prospects for practical applications.Finally, overall waveabsorbing materials are crucial for various applications, ranging from electromagnetic interference shielding to radar absorption.The prepared CNFO material offers several advantages over wave-absorbing materials, such as carbon materials, ceramics, conductive polymers, etc., as follows: (1) tunability: the ability to adjust the Co to Ni ratio in the CNFO allows for the netuning of the magnetic properties.This provides exibility in tailoring the material's performance for specic frequency ranges or applications.(2) Enhanced crystallinity: as evident from the sharp and narrow XRD peaks, the CNFO material possesses high crystallinity.This characteristic oen leads to improved electromagnetic properties and better stability.(3) High magnetic permeability: CNFO materials exhibit higher values of actual magnetic permeability, especially in the presence of increased Ni 2+ ions.Higher magnetic permeability is benecial for electromagnetic wave absorption as it facilitates the penetration of electromagnetic waves into the material, leading to increased absorption.(4) Balanced magnetic properties: the CNFO material balances the high magnetic moment of cobalt with the tunability provided by nickel.This balance ensures optimized absorption across a wide frequency range.

Author contributions
Marwa M. Husseininvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); data curation (management activities to annotate (produce metadata)); Samia A. Saafaninvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); data curation (management activities to annotate (produce metadata)).H. F. Abosheiashainvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); data curation (management activities to annotate (produce metadata)).Di Zhouinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); data curation (management activities to annotate (produce metadata)).D. S. Klygachinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); data curation (management activities to annotate (produce metadata)); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyze or synthesize study data).M. G. Vakhitovdata curation (management activities to annotate (produce metadata)); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyze or synthesize study data); S. V. Trukhanovinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); methodology (development or design of methodology; creation of models). A. V. Trukhanovinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); supervision (oversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core team); conceptualization (ideas; formulation or evolution of overarching research goals and aims); funding acquisition (acquisition of the nancial support for the project leading to this publication).T. I. Zubarinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyse or synthesize study data) and methodology (development or design of methodology; creation of models); K. A. Astapovichinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyse or synthesize study data); dra preparation.Hesham M. H. Zakalyinvestigation (conducting a research and investigation process, specically performing the experiments, or data/evidence collection); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyse or synthesize study data) and methodology (development or design of methodology; creation of models).Moustafa A. Darwishinvestigation (conducting a research and investigation process, specically performing the experiments, or data/ evidence collection); supervision (oversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core team); formal analysis (application of statistical, mathematical, computational, or other formal techniques to analyse or synthesize study data) and methodology (development or design of methodology; creation of models); funding acquisition (acquisition of the nancial support for the project leading to this publication).
−1 indicate the presence of additional O-H (or C-H/C-C) groups, conrming the existence of interlayer water and the oscillations of H-O-H bonds.The stretching mode of O-H bending vibration is attributed to the absorption band observed at 3411 cm −1 .
(a)-(c) present Transmission Electron Microscopy (TEM) micrographs of CoFe 2 O 4 captured in diffraction mode, while Fig. 4(d) and (e) depict High-Resolution Transmission Electron Microscopy (HRTEM) images.Fig. 4(f) demonstrates the Selected Area Electron Diffraction (SAED) pattern, and Fig. 4(g) illustrates a histogram representing the distribution of particle sizes.
Fig.6illustrates the frequency-dependent behavior of the permittivity for CNFO with (x = 0.0-1.0).The measurements were conducted within the frequency range of 8 to 18 gigahertz.The values of the real and imaginary parts of the electric permittivity and magnetic permeability were calculated from the measured S-parameters.In Fig.6a, one can analyze the behavior of the real part of the permittivity.It is shown that the x = 0.0 sample is characterized by the minimum values of the real part permittivity (∼1.65-1.68).It is noted that with increasing Ni 2+ concentration, the values of permeability increase monotonically and are: ∼1.71-1.73(for x = 0.3); ∼1.74-1.75(for x = 0.5); ∼1.77-1.78(for x = 0.7).The sample x = 1.0 (∼1.88-1.89)was characterized by the maximum value of the real part permittivity.It is shown that in the region of 16.3-17.5GHz,local minima are noted for all the samples.For both samples with x = 0.0 and 1.0, two clear minima are noted.While for the rest of the samples, there is blurring by local minima (oen with the formation of a wide plateau, as for x = 0.5).The amplitude of these minima correlates well with the nickel concentration.So, it can be noticed that the minimum value of the amplitude is typical for x = 0.0 (∼1.42), while the maximum value is noted for x = 1.0 (∼1.62).

Fig. 6
Fig. 6 Frequency dependences of the permittivity of CNFO (x = 0.0-1.0).(a) Real part of the permittivity; (b) imaginary part of permittivity; (c) mechanism of the electrical losses in condensed matter in the highfrequency range.

Fig. 7
Fig. 7 Frequency dependences of the permeability of CNFO (x = 0.0-1.0).(a) Real part of the permeability; (b) imaginary part of permeability; (c) mechanism of the magnetic losses in condensed matter in the high-frequency range.